Learn Russian: Я должен прийти домой в 18:50, чтобы посмотреть сериал. - I must arrive home at 18:50 to watch the TV series.

You read it as в восемнадцать пятьдесят. For exact digital times, Russian works just like English: you use the preposition в (at), followed by the cardinal number for the hour (восемнадцать), and then the number for the minutes (пятьдесят).

Would a female speaker say this exact same sentence?

Not quite! The word должен (must / have to) behaves like a short adjective and must match the gender of the person it describes. A woman would say Я должна прийти... instead.

Why is the word for home домой here instead of дома?

Домой implies direction ("homewards" or "to home"), answering the question "where to?". Because the action is arriving (прийти), there is motion involved towards a destination. You only use дома when you are already located at home.

How does чтобы work in this sentence?

Чтобы translates to "in order to" or just "to" when expressing a purpose. Because the same person is doing both actions (arriving and watching), you simply follow чтобы with the infinitive verb посмотреть.

Why do we use the perfective verbs прийти and посмотреть here?